Sampling and Wavelet

Question 1

Explain what happens to the spectrum of a signal when downsampling by two.

Answer 1

1. The spectrum of the sampled signal contains (scaled) copies of the original spectrum.
2. If the sampling rate is reduced, then the replicas are placed closer together in the signal spectrum.

Question 2

What is telecine? What artefacts can be caused by telecine?

Answer 2

This was when broadcasters were using cinema film cameras to record the TV screens.

1. There’s aliasing when fine patterns in the film being digitised without proper filtering. This causes wavy or shimmering interference patterns (high frequency areas).

Question 3

How is aliasing avoided in Digital Cameras?

Answer 3

1. Anti-aliasing filter in front of the digital camera.
2. Blurs the incoming light.
3. Blur spreads the light from one point over multiple sensor pixels.
4. Very high-frequency detail is softened just enough to prevent interference patterns.

Question 4

What do we mean by near-balanced filters? Name an example of un-balanced wavelet and show how these unbalanced filters might affect an application?

Answer 4

1. A wavelet filter pair where the low-pass and high-pass filters are symmetric or nearly symmetric, and their energy is evenly distributed.
2. Obtain more balanced system by using longer filters.

Question 5

Is the Discrete Wavelet Transform Shift Invariant and why?

Answer 5

A transform is shift invariant if shifting the input signal results in a corresponding shift in the output without changing the transform coefficients in structure or value.

Question 6

Give an example of a 1D signal, where subsampling by 2 could lead to aliasing.

Answer 6

Assume signal x(t) = sin(2pift)

1. Let f = 3 Hz
2. Let sampling freq = 8 Hz

Aliasing as N_f is 4 Hz and the frequency is only 3 Hz

Question 7

Explain why the concept of PR is key in the use of the DWT in image compression.

Answer 7

1. As in the reconstruction filter, it implements an upsampling feature that cancels the downsampling in the analysis filters.
2. This helps the DWT be shift-variant.

Question 8

Explain how the rate-distortion curves can be used to compare the performance of different sets of DWT filters in image compression systems.

Answer 8

1. Let’s say that the MSE is the distortion and the entropy is the rate.
2. Can see how the different transforms perform in entropy (no. of optimum bits per symbol) for a certain amount of MSE.

3.